Notes on Solute Partitioning and the Distribution Coefficient in Liquid Phases

11.4 The Partitioning of a Solute Among Two Coexisting Liquid Phases; The Distribution Coefficient

Overview of Solute Distribution Phenomenon

• When a gas, liquid, or solid is introduced into two partially miscible or completely immiscible solvents, it alters the solute distribution between the two liquid phases based on the solute amount present.

• The distribution of a solute among liquid phases is crucial in several fields:
    - Industrial: Important in purification processes like liquid extraction and partition chromatography.
    - Pharmacological: Relevant for understanding drug distribution between lipids and body fluids.
    - Environmental: Aiding in understanding pollutant distribution in different environmental media (air, water, soil).

Distribution Coefficient (K)

• The distribution coefficient (K) quantifies how a solute divides between two coexisting liquid phases.
    - Defined as:
    $K = \frac{\text{Concentration of solute in phase I}}{\text{Concentration of solute in phase II}}$
    - This section aims to:
      1. Predict the distribution coefficient for a solute if experimental data is lacking.
      2. Utilize experimental K values to determine liquid-phase activity coefficients.

Phase Behavior Considerations

• Focus on cases where the solute addition does not influence the miscibility of the solvents.
    - This covers scenarios where either the concentration of solute added is very small, or the solvents are sufficiently immiscible.

Equilibrium Relations in Solute Distribution

• The equilibrium distribution is determined via:
    $\mathrm{f}_{1}(T,P,x^{1}) = \mathrm{f}^{\prime}(T,P,x^{1})$

• Conservation of moles of solute:
    $N_{1} = N + N_{II}$
    - Where N is the number of moles in phase I.

• The activity coefficient for phase I is expressed as:
    $\mathrm{f}_{1}(T,P,x) = x(T,P)\mathrm{f}(T,P)$

Liquid-liquid Equilibrium Relation

• Rearranging gives:
    $K_{x} = \frac{(T,P)}{(T,P,x^{1})}$
    - This establishes that the distribution coefficient in terms of mole fractions relates inversely to the ratio of activity coefficients in the two phases:
    $K = \frac{\gamma_{1}}{\gamma_{2}}$

• Given activity coefficient data, one can compute solute distribution or derive activity coefficients from concentration distributions.

Example: Calculation of Activity Coefficients

Data for Bromine in Carbon Tetrachloride and Water at 25°C

• 
  
  

• Concentrations and distribution coefficients (K):
  
  

  
  

  Concentration of Bromine (kmol Br2 in CCl4)	Kc
  0.04	26.8
  0.1	27.2
  0.5	29.0
  1.0	30.4
  1.5	31.4
  2.0	33.4
  2.5	35.0
  Pure Species Data

  Species	Molecular Weight	Density (kg/m³)
  CCl4	153.84	1.595 x 10³
  H₂O	18.0	1.0 x 10³
  Br2	159.83	3.119 x 10³

  Calculation Process

  • Determine the mole fractions for bromine (using its known concentration) in both phases:
      1. Carbon Tetrachloride (CCl4):
         $X_{Br_2,CCl4} = \frac{C_B}{C_{CCl4}}$
      2. Water (H₂O):
         $X_{Br_2,H_2O} = \frac{C_B/K_c}{C_{H_2O}}$

  • Mole fractions will inform calculations for activity coefficients.

  Situational Analysis: Undissolved Solute in Two Immiscible Solvents

  • Consideration of a scenario with undissolved solute in equilibrium with liquid phases:

  • The relationship holds:
      $\mathrm{f}_{1}(T,P) = \mathrm{f}^{\prime}(T,P,x^{1})$

  Example - Partitioning in Water-N-Octanol System

  • Mixture allows for chemical partitioning based on hydrophilic/hydrophobic characteristics:
    $K_{ow,i} = \frac{\text{Concentration of species } i \text{ in octanol}}{\text{Concentration of species } i \text{ in water}}$

  • Organic chemicals tend to exhibit large octanol-water partition coefficients, indicating a preference for the octanol phase.

  Octanol-Water Partition Coefficient Measurement Challenges

  • Measurements may yield inaccuracies due to the low concentration of hydrophobic compounds in aqueous solutions.

  • A rough estimate utilizes the density and molecular weight metrics for the respective solvent:
      $CO \approx \frac{0.827 ext{ g/cc}}{130.22 ext{ g/mol}} ext{ compared to } \frac{1 ext{ g/cc}}{18 ext{ g/mol}}$

  Regression and Correlation Equations

  • Correlations proposed for the estimation between computing activity coefficients, partition coefficients, and solubility dynamics:
      - From infinite dilution activity coefficients, approximate partitions using:
        $\log_{10} K_{ow,i} = -0.486 + 0.806 \log_{10} Y_{W,\infty}$

  Case Studies

  1. Purification Process for Benzylpenicillin
        - Uses data of concentrations and partition coefficients through a mass balance approach.
        - Establish necessary expressions for evaluating concentrations in both phases post-equilibrium.

  2. Gibbs Energy of Transfer Calculations
        - Analyze solvation free energies when transferring species between two liquid phases. Favorable/Unfavorable transfers:
          - $\Delta G < 0$ indicates a favorable process where energy is released.      - $\Delta G > 0$ implies energy required for transfer.